Download Division 3AA/4AA - ICTM Math Contest

WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
ALGEBRA I
PAGE 1 OF 3
1.
The circumference of Circle O is 34, the circumference of Circle P is 42, and the
circumference of Circle T is 53. If one of the three circles is selected at random, find the
probability that the radius of that circle is more than 6. Express your answer as a
common fraction reduced to lowest terms.
2.
The sum of two numbers is 106, and their difference is 18. Find the larger of the two
numbers.
3.
If 17.4 x  34.2  5.8  3 x  y  , find the value of y . Express your answer as an improper
fraction reduced to lowest terms.
4.
For all values of x such that x  3 ,
5 x  15
4 x  12

 k . Find the value of k .
12
15
Express your answer as a decimal.
5.
If the sum of two numbers is 15 and the product of the two numbers is 5, find the sum of
the reciprocals of the two numbers.
6.
The repeating decimal 0. 522 (where the 522 repeats) is expressed as a common fraction
reduced to lowest terms. Find that common fraction.
7.
Three different fruits lie on a table: an orange, an apple, and a banana. If two different
fruits are selected at random from among these three, find the probability that the first
letter of each fruit selected is a vowel. Express your answer as a common fraction
reduced to lowest terms.
8.
Assume that men and women are adults, and assume that boys and girls are not adults.
At a Halloween party, only men, women, boys, and girls attended. There were 17 girls,
11 adults without costumes, 14 women, 10 girls with costumes, 29 people without
costumes, 11 women with costumes, and 16 males with costumes. Find the total number
of people who attended this Halloween party.
9.
Emily travels at a rate of 4k mph. for 4 miles, 5k mph. for 5 miles, etc., and finally at
the rate of wk mph. for w miles. For what value of w will the total distance traveled
divided by the total time it took (average rate) be 12 times her starting rate?
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
ALGEBRA I
PAGE 2 OF 3
10.
When Sam started to work at Harriet’s Hamburgers, he asked Harriet what his hourly pay
would be. Harriet said Sam could pick any positive dollar amount he wanted (such as
$17.42), and then Harriet would subtract the square (percentage-wise) of that amount
from twice his picked dollar amount to arrive at his net pay per hour. For example, if
Sam picked $4.00, Harriet would subtract (4.00) 2 per cent of $4.00 from $8.00, and
Sam’s net pay per hour would be $7.36. Find the amount Sam should pick in order to
maximize his net hourly pay. After you have found that amount, round that amount to
the nearest cent. Then express your answer in dollars and cents with, of course, the
cents rounded to the nearest cent.
11.
If
12.
When
x  600  600  600  600   , find the value of x .
a 1b 1  ab
is expressed in simplest form with positive exponents, the result is
ab 1  (ab) 1
1  ak bw
where k , w , p , and g are positive integers. Find the smallest possible value
ap  g
of  k  w  p  g  .
4kx
where k is an integer and where x  0 . If f  f ( x)   x for all x  0 ,
3x  4
then find the value of k .
13.
Let f ( x) 
14.
(Multiple Choice) For your answer write the capital letter which corresponds to the
correct choice.
If 5 x  7 y  z , and x , y , and z are positive integers, then all of the following must be
an integer except:
A)
z
xy
B)
z
7
C)
z
35
D)
z
5
E)
y
5
Note: Be certain to write the correct capital letter as your answer.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
ALGEBRA I
PAGE 3 OF 3
15.
Let N be a four-digit number (with the thousands digit non-zero) such that the
units digit of N 2 is 9, and such that the sum of the digits of N is 10. How many distinct
possibilities exist for the value of N ?
16.
Let x represent a positive number base such that 129 x  233ten . Find the value of x .
Express your answer in base ten.
17.
Candle A is 5 times as long as Candle B. Candle A is consumed uniformly in 4 hours,
and Candle B is consumed uniformly in 6 hours. If the 2 candles are lit at the same time,
find the absolute value of the difference between the number of hours it will take for
Candle A to be 3 times as long as Candle B and the number of hours it will take for
Candle B to be 3 times as long as Candle A. Express your answer as an improper
fraction reduced to lowest terms.
18.

 y 

y  3 x

 
If x  2 and y  3 , find the value of x
. Express your answer as an exact
integer.
19.
Two positive integers each of which is less than 10 form a two-digit number. The digits
are then switched to form a second two-digit number. When the second two-digit
number is subtracted from the original two-digit number, the result is a number that is
one less than the second two-digit number. Find the original number.
20.
Serena was rowing upstream one day when her cap blew off into the stream. She failed
to notice it was missing until 18 minutes after it blew off. She immediately turned
around and recovered the cap 2.64 miles downstream from where it initially blew off.
Assume Serena’s physical rate of rowing was constant, the rate of the current was
constant, and that it took no time to turn around. Find the number of miles per hour in
the rate of the current. Express your answer as an exact decimal.
2012 RAA
Algebra I
Name
ANSWERS
School
(Use full school name – no abbreviations)
Correct X
2 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
2 (Must be this reduced
1.
2.
3.
4.
5.
common fraction.)
3
11.
62
171
29
12.
7
13.
1
(Must be this reduced
improper fraction.)
1.5625
(Must be this
decimal.)
3
625
A
14.
15.
34
(Must be this reduced
common fraction.)
6.
58
111
(Must be this reduced
common fraction.)
7.
1
3
8.
66
18.
32768
9.
92
19.
73
10.
8.16
(Must be this
capital letter.)
16.
17.
(Must be in dollars
and cents, $ optional.)
14 OR 14ten OR 1410
160
129
4.4
20.
(Must be this reduced
improper fraction, hours optional.)
(Must be this decimal,
mph optional.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
GEOMETRY
PAGE 1 OF 3
1.
Triangle ABC has sides of lengths 6, 8, and 10. Triangle DEF has sides of lengths 5,
12, and 13. Triangle GHJ has sides of lengths 3.5, 12, and 12.5. If one of these
triangles is selected at random, find the probability that the triangle selected has an area
greater than 25. Express your answer as a common fraction reduced to lowest terms.
2.
In ABC , if AB  BC , and BAC  32 , by how many degrees
does the angle measure of ABC exceed that of BCA ?
3.
(Multiple Choice) For your answer write the capital letter
that corresponds to the best answer.
B
A
The midpoint of the hypotenuse of a right triangle is what center of the triangle?
A) Circumcenter
B) Incenter
C) Center of gravity
D) Orthocenter
Note: Be sure to write the correct capital letter as your answer.
4.
Two triangles have congruent bases. The heights of each of these two triangles on these
two congruent bases are also congruent. Must these two triangles be congruent? For
your answer, write the whole word Yes or No—whichever is correct.
5.
(Always, Sometimes, or Never True) For your answer, write the whole word Always,
Sometimes, or Never—whichever is correct.
If the slope of a line is a positive number, then the line does not pass through (intersect)
Quadrant IV.
6.
In rectangle RECT , M is the midpoint of RE , N is the midpoint of EC , O is the
midpoint of TC , and P is the midpoint of RT . If RP  6 , and TO  17.5 , find the
perimeter of quadrilateral MNOP .
7.
From all the interior angles of an equiangular pentagon, an equiangular hexagon, and an
equiangular heptagon, one of these interior angles is selected at random. Find the
probability that the angle selected has a degree measure that is an integer. Express your
answer as a common fraction reduced to lowest terms.
C
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
8.
GEOMETRY
PAGE 2 OF 3
In the diagram, ABCD is an isosceles trapezoid with BC as one of
the bases. AD  kx  4 , DC  7.6 x  5.1 , BC  19 x , and
AB  3.2 x  21.3 . If k is a positive integer, find the smallest
possible value of k .
B
A
D
9.
10.
11.
C
In the diagram, a  b with a non-perpendicular transversal. If
two of the numbered angles are selected at random, find the
probability that the two angles selected are congruent. Express
your answer as a common fraction reduced to lowest terms.
3
1 2
a
Given the points D 1, 2  , E 10, 2  , and F  9, 2  . A lattice
point is defined as a point for which all coordinates are integers. Find the number of
distinct lattice points that are in the interior of DEF.
In the diagram, OMP  RMP . MO  8 , OP  6 , and
MR  12 . O , P , and R are collinear. The area of MOR
w k
can be expressed as
. If k and w are positive
4
integers, find the smallest possible value of  k  w  .
4
b
M
O
P
12.
If 4320 ' were expressed as an improper fraction in terms of degrees, the answer would
130
2
be
(° optional). By how much does 64  exceed 6025'30" ? Express your
3
3
answer as an improper fraction in terms of degrees.
13.
AM , BN , and CP are the respective medians of Triangle ABC . If AM  37.02 ,
BN  33.18 , and CP  39.51 , find the length of MN . Express your answer in decimal
form rounded to 4 significant digits.
R
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
GEOMETRY
PAGE 3 OF 3
14.
A rectangular solid has dimensions of 5, 6, and 8. From one of the faces with smallest
area, a circular hole of diameter 4 is drilled at a right angle from the face halfway through
to the opposite face. When the circular hole reaches that halfway point, a semi-circular
hole of diameter 4 is drilled the rest of the way to the opposite face and at right angles
with that opposite face. Find the total surface area of the rectangular solid with the hole.
Express your answer as a decimal rounded to the nearest tenth.
15.
(Always, Sometimes, or Never True) For your answer, write the whole word Always,
Sometimes, or Never—whichever is correct.
If the hypotenuse of a right triangle is parallel to one of the axes, then the product of the
slopes of the two legs is 1 .
16.
A circle with center at O is inscribed in ABC . AB  12 ,
BC  5 , and AC  13 . Point E lies on BC , and point D lies
 

on AB such that DE  AC and such that DE passes
through point O. Find DE . Express your answer as an
improper fraction reduced to lowest terms.
C
E
O
B
D
17.
The circle whose equation is x 2  y 2  6 x  10 y  18.2729 is tangent to the line x  k
where k  0 . Find the value of k . Express your answer as a decimal.
18.
The number of degrees in the sum of the degree measures of the interior angles of a
convex polygon is 15858 more than the number of distinct diagonals that can be drawn in
that polygon. If the number of sides of the polygon is not an integral multiple of 17, find
the number of sides of the polygon.
19.
If the area of a circle is 196 , the length of a 144 arc can be expressed as k . Find the
value of k . Express your answer as a decimal.
20.
In ABC , the vertices are at A  4,3 , B 14,5  , and C  3,1 . Find the length of the
median from C to AB .
A
2012 RAA
Geometry
Name
ANSWERS
School
(Use full school name – no abbreviations)
Correct X
2 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1 (Must be this reduced
1.
2.
3
5.
6.
7.
A
NO
Sometimes
10.
(Degrees optiona.)
12.
(Must be this
capital letter.)
13.
(Must be the
whole word.)
14.
(Must be the
whole word.)
74
11
18
1
3
15.
16.
(Must be this reduced
common fraction.)
17.
5
8.
9.
11.
84
3.
4.
common fraction.)
18.
(Must be this reduced
common fraction.)
12
19.
20.
1468
509
120
(Must be this reduced improper
fraction, degrees optional.)
19.38
314.8
Always
221
30
(Must be this
decimal.)
(Must be this
decimal.)
(Must be the
whole word.)
(Must be this reduced
improper fraction.)
10.23
(Must be this
decimal.)
159
11.2
3 17
(Must be this
decimal.)
(Must be this
exact answer.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
1.
ALGEBRA II
PAGE 1 of 3
From the set 2,5,8 , one member is selected at random and substituted for x in the
expression 3 x  8 . Find the probability that after the substitution is made, the expression
will have a value greater than 20. Express your answer as a common fraction reduced to
lowest terms.
2.
A piece of property is assessed at $46,000. The tax rate for a certain year on the assessed
property value is $2.50 per $1000. Find the number of dollars of tax paid on this piece of
property for that certain year.
3.
For all real values of x except for x   7 ,
2 x5  x 4  3x3  x  5
kx  w
. Find the value of  k  w  .
 2 x3  x 2  17 x  7  2
2
x 7
x 7
4.
Clerk A, who sorts at a constant rate, sorts B letters per hour. Clerk C, who sorts at a
constant rate, sorts D letters per hour. Every ten minutes, the number of letters sorted by
Clerk A exceeds the number of letters sorted by Clerk C by 10. During a certain day
each of these two clerks sorted letters for k hours. For that day, Clerk A sorted 420 more
letters than Clerk C. Find the value of k .
5.
If x 4  3 x3  8 x 2  2 x  12 is factored with respect to the integers, one of the factors is
x3  kx 2  wx  f . Find the value of  k  w  f  .
6.
On a flat planar surface, a semicircle is drawn in the exterior of a square with a side of the
square as its diameter. If the sum of the areas of the square and the semicircle is 240,
find the length of the diameter. Express your answer as a decimal rounded to the nearest
hundredth.
7.
Four circles have radii of respective lengths 5.13 , 2.15 , 7.86 , and 4.57 . If two of the
circles are selected at random without replacement, find the probability that each of the
circles selected has an area greater than 61.65. Express your answer as a common
fraction reduced to lowest terms.
8.
The equation of the hyperbola whose foci are at  6, 0  and  6, 0  and which has a latus
rectum whose length is 18 can be expressed in the form
 3k  2w  .
x2
y2

 1 . Find the value of
k
w
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
9.
ALGEBRA II
PAGE 2 of 3
Let f  x   x 2  5 and let g  x   kx  w where k and w are positive integers. If
f  g  6    1686 , find the possible values of  k  w  . For your answer, give the smallest
possible value of  k  w  .
10.
11.
Let 7,8, 4, 9,3, 21, 59, be a sequence of numbers such that each present term after the
fifth term is 3 more than twice the sum of the preceding term and the number of the
present term. If n is the number of the term that equals 671088573, find the value of n .
The points A 1, 3 , B  4, 2  , and C  8,16  are the vertices of ABC . The length of the
longest altitude of ABC can be expressed as k w , where k and w are positive
integers. Find the smallest possible value of  k  w  .
12.
Let i  1 . If the reciprocal of 7  2i is written in x  yi form where x and y are real
numbers, find the value of  4 x  2 y  . Express your answer as a common fraction
reduced to lowest terms.
13.
Let k and w be real numbers such that x 4  10ix 3  kx 2  wix  296  0 where i  1 .
If one of the solutions for x of the given 4th degree equation is 7  5i and at least two of
the solutions for x of the given 4th degree equation are real numbers, find the value of w .
14.
Find the 50th term of the arithmetic sequence 5,
24 23
, ,  . Express your answer as a
5 5
decimal.
15.
The equation of the graph which represents the locus of all points P in a plane such that
the ratio of the absolute value of its distance from the point  0, 4.5  to the absolute value
of its distance from the line y 
the value of  k  w  .
1
is 9 can be expressed in the form ky 2  x 2  w . Find
18
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
16.
ALGEBRA II
PAGE 3 of 3
Let i  1 . If x and y are real numbers, find the value of  x  y  when
1
3x  6 yi  18  2
1
0
0
3 i
3 6
1
2
3
17.
A positive integer, if divided by 12, leaves a remainder of 11; if divided by 11, leaves a
remainder of 10; if divided by 10, leaves a remainder of 9; if divided by 9, leaves a
remainder of 8; if divided by 8, leaves a remainder of 7; if divided by 7, leaves a
remainder of 6; if divided by 6, leaves a remainder of 5; and if divided by 5, leaves a
remainder of 4. Find the smallest such positive integer.
18.
If x represents a positive integer, find the sum of all distinct values of x such that
x  x  4  x  2   0 .
19.
Find the value of
  n  7
12
n 5
20.
2
.
Let ABC be a right triangle with ACB  90 , AC  6 , and BC  2 . E is the midpoint
of AC , and F is the midpoint of AB . If CF and BE intersect at G , then cos(CGB) ,
in simplest radical form, is
value of  k  w  f  .
k w
where k , w , and f are positive integers. Find the
f
2012 RAA
Algebra II
Name
ANSWERS
School
(Use full school name – no abbreviations)
Correct X
2 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
2 (Must be this reduced
1.
common fraction.)
3
11.
2.
115
3.
172
6.
7.
12.
13.
7
4.
5.
($ or dollars
optional.)
14.
13
13.13
1
2
15.
(Must be this
decimal.)
16.
(Must be this reduced
common fraction.)
15
32
53
(Must be this reduced
common fraction.)
40
4.8
100
10
17.
27719
8.
81
18.
10
9.
11
19.
60
10.
30
20.
267
(Must be this
decimal.)
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
1.
2.
PRECALCULUS
PAGE 1 OF 3
In Bag A are 2 red marbles, 2 green marbles, and 2 white marbles. In Bag B, there are 2
red marbles and 1 green marble. Martha selects one of the marbles at random from Bag
B and places this marble in Bag A. After Martha places this marble in Bag A, find the
probability that there are more red marbles in Bag A than there are green marbles in Bag
A. Express your answer as a common fraction reduced to lowest terms.
The distance between two points represented by  x,5  and  3, 26  is 29. Find the
largest possible value of x .
3.
In which quadrant is cotangent negative and cosine positive? Express your answer as a
Roman numeral.
4.
Let i  1 . The two non-real roots of x 4  x 2  12  0 are wi and ki . Find w  k .
5.


If the vector u   4, 7,1 and the vector v   2, 3, 2 , then find the magnitude of the

sum vector u  v .
6.
The graph of y 
7.
Let i  1 . If x and y are real numbers such that 3x  5  7 yi  2 y  3 xi , find the
value of y .
8.
In Triangle ABC , altitude AH and median BM intersect inside the triangle and have
equal lengths. If ABM  45 and AB  16 , then, in simplest radical form, the length of
AH is k w  f p where k , w , f , and p are positive integers. Find the value of
2 x3  7 x 2  5 x  6
has an oblique or slant asymptote of y  kx  w .
x2  5x  6
Find the value of  k  w  .
k  w  f
 p .
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
9.
10.
11.
12.
13.
PRECALCULUS
PAGE 2 OF 3
From a 20" 32" rectangular sheet of cardboard, x " x " squares are cut from two
corners, and x " 16" rectangles are cut from the other 2 corners. The remaining
cardboard is then folded to form a box with a lid. Find the value of x that yields the box
of maximum volume. Express your answer as a decimal rounded to the nearest
hundredth of an inch.
x 4  5 x3  2 x 2  11x  1
The graph of f ( x) 
has vertical asymptotes at x  k and at
x 2  10 x  3
x  w . If k  w , find the value of k . Express your answer as a decimal rounded to the
nearest ten-thousandth.
Cindy has three marbles—one is red, one is white, and one is green. She has two
containers—A and B. She selects a marble at random and then chooses a container at
random into which she puts that marble. She then repeats this process for each of the two
remaining marbles. Find the probability that the red marble was the first marble selected
and that the white marble is in the same container as the red marble. Express your
answer as a common fraction reduced to lowest terms.
 2  6  
4
12
24
k where k is a positive integer. Find the value of k .
In the diagram (not drawn to scale), Points A , B , C , D , and E
  
lie on a circle with center at F . BC  AD  FE ,
A
AB  BC  DE , and FE  8 . Find the exact value of  AD  DE  .
B
F
C
D
14.
E
A student received a grade of 76 on a final examination in math for
which the mean grade was 70 and the standard deviation was 5.
Find the “standard score” or “z-score” on this exam for this student. Express your answer
as a decimal.
WRITTEN AREA COMPETITION
ICTM REGIONAL 2012 DIVISION AA
15.
16.
PRECALCULUS
PAGE 3 OF 3
A right circular cylinder has a total surface area of 154 . Find the length of the radius
that is needed for the right circular cylinder to have maximum volume. Express your
answer as a decimal rounded to the nearest hundredth.
In ABC , AB  7 , and AC  8 . cos  BAC  is a rational number that can be expressed
k
where k and w are relatively prime positive integers. It is known that  k  w  is
w
the cube of a positive integer and that BC is a positive integer. Find the sum of all
possible distinct values of BC .
as
17.
If ( x  y )5 is expanded and completely simplified, find the sum of the numerical
coefficients.
18.
If i  1 , find 21  72i .
19.
In the diagram, a circle with center at P is externally tangent to the

circle with center at O . Circle P is also tangent to OA at A and

to OC at C . AOC  26 , and the length of a radius of the circle
with center at P is 2.416 . Find the length of a radius of the circle
with center at O . Express your answer as a decimal rounded to the
nearest thousandth.
20.
O
A
P
C
The Polar Coordinates of points D , E , and F are D  42,154  , E  93,117  , and
F  67,1  . Find the radius of the inscribed circle of Triangle DEF . Round your answer
for the radius to the nearest integer and express your answer as that integer.
2012 RAA
Name
Pre-Calculus
ANSWERS
School
(Use full school name – no abbreviations)
Correct X
2 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
(Must be this reduced
2 (Must be this reduced
1
1.
2.
3.
4.
5.
common fraction.)
3
17
IV
2 3
137
7.
8.
9.
10.
12.
2304
13.
(Must be this
exact answer.)
14.
(Must be this
exact answer.)
1
16
4.00
11.
(Must be this
Roman numeral.)
5
6.
8
1.2
15.
5.07
16.
19
17.
32
18.
75
(Must be this decimal,
inches optional.)
0.3096 OR .3096
common fraction.)
6
19.
(Must be
this decimal.)
20.
8.324
22
(Must be this
decimal.)
(Must be this
decimal.)
(Must be this
decimal.)
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 1 of 4
NO CALCULATORS
1.
From the four names—Truman, Ryan, Daley, and Bush—1 name is selected at random.
Find the probability that the name selected was composed of exactly four letters. Express
your answer as a common fraction reduced to lowest terms.
2.
(Always, Sometimes, or Never True) For your answer, write the whole word Always,
Sometimes, or Never—whichever is correct.
The median of a triangle divides the triangle into two triangles of equal area.
3.
If x is an integer, find the sum of all distinct values of x such that x  3  5 .
4.
Given four triangles with side-lengths of each triangle as shown by the ordered triples:
 3, 4,5 ,  5,12,14  , 
 
43, 53, 4 6 ,

7,1,3 . If one of the four triangles is selected
at random, find the probability that the triangle selected is a right triangle. Express your
answer as a common fraction reduced to lowest terms.
B
A
5.
In the diagram, AB is tangent to the circle at B . Points A ,
C , and D are collinear, and C and D lie on the circle. If
AC  3 and AB  78 , find CD .
C
D
6.
A line contains the point  2, 2  and is perpendicular to a second line that contains the
points  5, 1 and  2,8  . The equation of the first line can be expressed as
x  ky  w  0 . Find the value of  2k  3w  .
7.
Given the 4 points: A  2, 4  , B  20, 7  , C 10,17  , D 18,12  . Find the area of the region
that is in the exterior of ACD but is in the interior of ABC .
8.
The perimeter of a scalene triangle is 22, and the longest side has a length of 10. Find
the longest possible length of the shortest side if all sides have lengths that are whole
numbers.
NO CALCULATORS
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 2 of 4
NO CALCULATORS
9.
On a plane surface, two secants of a circle intersect at point P in the exterior of the
2
circle. The angle formed by the two secants intercepts arcs which are respectively
and
5
1
of the circle. Find the degree measure of the acute angle formed by these two secants.
4
10.
Each of the three circles shown in the diagram is tangent to the
other two. If the radii of the three circles have lengths of 1, 2, and
9, find the exact area of the triangle formed by connecting the
centers of the three circles.
11.
3 5 a b c

where a , b , c , and d are positive integers. Find the smallest
d
3 5
possible value of  a  b  c  d  .
12.
The sides of a triangle have lengths of 5, 7, and 8. The area of the circumscribed circle of
k
this triangle can be expressed as
where k and w are positive integers. Find the
w
smallest possible value of  k  w  .
13.
There are three consecutive odd integers. If the largest is multiplied by 4 and then the
result is added to the smallest, the result is 181. Find the smallest of the three
consecutive odd integers.
14.
From five angles—two of which are acute, one of which is right, and two of which are
obtuse, two angles are selected at random without replacement. Find the probability that
neither of the angles selected has a cosine which is negative. Express your answer as a
common fraction reduced to lowest terms.
NO CALCULATORS
FROSH-SOPH EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 3 of 4
NO CALCULATORS
15.
  
In the diagram, DC  AB , BD bisects ADC and ABC ,
and points A , B , and E are collinear. CBE  60 , and
ABCD has a perimeter of 240. Find the perimeter of ABD .
D
A
C
E
B
16.
If m and n are the two distinct roots of x 2  154 x  5  0 , find the value of (m  n) 2 .
17.
In a different galaxy, two planets revolve around their sun in
concentric square orbits. Planet A travels in a clockwise direction at
2 million miles per hour in an orbit of length 8 million miles. Planet
B travels in a clockwise direction at 5 million miles per hour in an
orbit of length 16 million miles. Initially, Planet A has a 45° head
start (see diagram). Find the number of hours it will be before the
two planets are, for the first time, separated by the maximum possible
distance.
B
A
S
mBSA=45
18.
Use each of the four digits 2, 7, 8, and 9 exactly once to form the numbers k and w in
the equation k  w  10 . Find the value of the product  kw  .
19.
The length of an edge of a regular tetrahedron is 60. The absolute value of the difference
between the surface area of the circumscribed sphere of this tetrahedron and the surface
area of the inscribed sphere of this tetrahedron is k . Find the value of k .
20.
In the diagram, AEC is acute, B lies on AC , D lies
on EC , and AD and BE intersect at F . If
DC  4  DE  and AB : BC  3 : 2 , then the ratio of the
A
B
area of EFD to the area of quadrilateral BCDF is
k : w where k and w are positive integers. Find the
smallest possible value of  k  w  .
F
E
NO CALCULATORS
D
C
2012 RAA
School
Fr/So 8 Person
Correct X
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
1 (Must be this reduced
1.
common fraction.)
2
Always
33
4.
5.
6.
7.
8.
9.
10.
1
2
17
12.
52
(Must be this
whole word.)
2.
3.
11.
13.
(Must be this reduced
common fraction.)
14.
23
15.
18
16.
42
17.
5
27
6 3
18.
(Degrees
optional.)
19.
(Must be this
exact answer.)
20.
33
3
10
(Must be this reduced
common fraction.)
180
23736
10
2023
4800
17
(Hours
optional.)
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 1 OF 3
NO CALCULATORS
1.
2.
Let A  1, 2,3 . If two distinct members of A are selected at random, find the
probability that the sum of those two distinct members is 5. Express your answer as a
common fraction reduced to lowest terms.
Find the value of
log(128)
. Express your answer as a common fraction reduced to
log(256)
lowest terms.
3.
(Multiple Choice) For your answer write the capital letter that corresponds to the
best answer.
Define the difference set A  B as the set {x : x  A and x  B} .
A) B  A
B) B ' A '
C) A  B
D) A  B
 A  B 
E) None of the previous.
Note: B ' is defined to mean the complement of set B .
Note: Be sure to write the correct capital letter as your answer.
4.
At Urbana, 33 teams entered the single elimination basketball tournament in which a
team is eliminated after it loses one game. At Champaign, 13 teams entered the double
elimination basketball tournament in which a team is eliminated after it loses two games.
If there will be a champion after k games in Urbana and after a maximum of w games
in Champaign, find the value of  k  w  .
5.
Two fair, standard cubical dice are rolled one at a time. If the number on the uppermost
face of the first die rolled is a five, find the probability that sum of the numbers on the
two uppermost faces is more than eight. Express your answer as a common fraction
reduced to lowest terms.
6.
Find the sum of the first seventeen terms of an arithmetic sequence whose ninth term is 3
and whose seventeenth term is 27.
7.
(Multiple Choice) For your answer write the capital letter that corresponds to the
best answer.
For statements x and y , the implication x  y is logically equivalent to which of the
following?
A) y '  x '
B) y '  x
C) y  x
D) x '  y '
E) x '  y
Note: x ' denotes the negation of statement x .
Note: Be sure to write the correct capital letter as your answer.
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 2 OF 3
NO CALCULATORS
8.
The coordinate axes of the graph of the equation
9 x 2  3 xy  5 y 2  267  2 x 2  xy  9 y 2 are rotated through a positive acute angle ( ) so
k w
where k , w ,
f
and f are positive integers. Find the smallest possible value of  k  w  f  .
as to eliminate the xy term. The value of sin( ) can be expressed as
9.
It is known that x varies directly as y and inversely as the square of z . If x  5 when
y  100 and z  2 , find the value of x when y  100 and z  10 . Express your answer
as a common fraction reduced to lowest teems.
10.
Let k be the square of an integer. Let k have exactly 8, exactly 9, or exactly 15 distinct
positive integral divisors. If k  261 , find the sum of all distinct values of k .
11.
1 2 
 4 1 0 
   7 12  , find the value of k .
If 
3
4

 2 2 


2 k 1 


5 6 
12.
The range of the real-valued function y  3  x 
3  x is given by  y : k  y  w .
Find the value of  k  w  .
13.
The successive class marks (or class midpoints) in a frequency distribution of the weights
of students are: 123, 135, 147, 159, 171, 183, 195, and 207. The third lowest class of
weights ranges from k to w where k and w are positive integers. Find the value of
 k  w .
14.
Bob is a gambler. Two orange and four blue marbles are placed in a container. From
these six marbles, two are drawn at random without replacement. Bob is a winner if both
marbles are blue or if one marble is orange and the other marble is blue. Find the
probability that Bob is a winner. Express your answer as a common fraction reduced to
lowest terms.
JUNIOR-SENIOR EIGHT PERSON TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 3 OF 3
NO CALCULATORS
15.
16.
A ball is dropped from a height of 54 inches and strikes a flat surface. Each time the ball
2
strikes the flat surface, it rebounds to
of its previous height. Find the total number of
3
inches the ball will travel before it eventually comes to rest.
1
Find the radian period of the graph of y  2 tan(  ) .
4
17.
Let x  3a  2b , where a and b are chosen independently (i.e., with replacement) from
the twenty-five integers from 1 through 25 inclusive (with each integer having an equal
likelihood of being chosen). Find the probability that x is an integral multiple of 5.
Express your answer as a common fraction reduced to lowest terms.
18.
One card has blue on each side, and a second card has orange on one side and blue on the
other. One of these cards is selected at random and placed on a table. If the shown side
of the card on the table is blue, find the probability that the other side of that card is also
blue. Express your answer as a common fraction reduced to lowest terms.
19.
All ages in this problem are in years. Dick is one and a half times as old as he was when
he was one and a half times as old as he was when he was five years younger than half as
old as he is now. Find the number of years in Dick’s present age.
20.
In the coplanar diagram, points A , B , and C are collinear, AD  16 ,
and AC  21 . ADE  CDE , AED  90 , DAB  60 , and
AB  BC . Find BE . Express your answer as a decimal.
A
D
E
B
C
2012 RAA
School
Jr/Sr 8 Person
Correct X
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
(Must be this reduced
1.
1
3
(Must be this reduced
common fraction.)
2.
7
8
4.
5.
6.
11.
12.
B
3.
(Must be this
capital letter.)
13.
57
1
2
14.
(Must be this reduced
common fraction.)
15.
9.
10.
14
15
(Must be this reduced
common fraction.)
270
(Inches
optional.)
(Radians
optional.)
(Must be this reduced
common fraction.)
17.
157
625
(Must be this reduced
common fraction.)
18.
2
3
(Must be this
capital letter.)
12
(Must be this reduced
common fraction.)
957
294
16.
7.
1
5
6  2 3 OR 2 3  6
4
51
A
8.
3
common fraction.)
19.
20.
90
1.5
(Years
optional.)
(Must be this
decimal.)
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 1 of 3
1.
If g ( x)  x 6  x5  x 4  24.54 , find the value of g (3.111) .
2.
A regular polygon with 37 sides is inscribed in a circle with a diameter whose length is
15.47 . Find the area of this regular polygon.
3.
 x, y  is the solution for the following system,:
value of  x  y 
4.
5.
 6.25 x  3.48 y  33.87804
. Find the

12.96 x  4.32 y  11.14992
The length of an arc of a circle is 123.4 . If the radius of the circle is 36.2 , find the
degree measure of the central angle subtended by the arc.
Quadrilateral QUAD is inscribed in a circle. DQU   243 x  13  ,
QUA   517 x  186   , and UAD  113 x  7   . Find the degree measure of ADQ .
Express your answer in the form k w ' with the value of w rounded to the nearest
minute.
6.
7.
A triangle has a base of length 45.12 and a height on that base of length h . To form a
new triangle, the old triangle has its base of length 45.12 decreased by w and its height,
on that base, is the length h increased by 984.1 . If the area of the old triangle is the
p
same as the area of the new triangle, then w 
. Find the value of  p  q  . Express
hq
your answer as an exact decimal. Do not use scientific notation.
If the vector 1.384, 2.841,3.856  is perpendicular to the vector  2.845,5.997, k  , find
the value of k .
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
8.
PAGE 2 of 3
A graph of a curve of the form y  ax 3  bx 2  cx  d with a , b , c , and d rational
numbers, passes through the following points with  x, y  coordinates:  20, 16  ,
 8,12  , 10,3
and  28,16  . Let  k , w  be an ordered pair of the form  x, y  that lies
on the given curve. If k and w are both positive integers and if 29  k  99 , find the
sum of all possible distinct values of w . Express your answer as a whole number.
9.
If x  34.53 , find the value of x 2  x  2  x 1.584 . Express your answer as a decimal
rounded to the nearest thousandth. Do not use scientific notation.
10.
In Triangle ABC , AB  245 , and ABC  60 . If all sides of Triangle ABC have
integral lengths, find the smallest possible perimeter of Triangle ABC such that the
perimeter is greater than 1560. Express your answer for the perimeter as an exact
integer.
11.
One diagonal of a rectangle has a length of 48.42. Find the largest possible area of any
such rectangle.
12.
A six-sided die is weighted so that the probability distribution is as shown:
x
1 2 3 4 5 6
. The value, denoted by x on the table, of a single roll
1 1 1 1 1
P  x
k
3 4 12 6 12
of a die is defined to be the number that appears on the top of the die when it comes to
rest. Find the expected value on a single roll of a die. Express your answer as an
improper fraction reduced to lowest terms.
13.
Rounded to 5 decimal places, the total population standard deviation ( X ) of a set of k
numbers is 2.47861 . Rounded to 5 decimal places, the sample population standard
deviation  S x  of the same set of k numbers is 2.48195 . Naturally, k is a positive
integer. Find the value of k . Express your answer as an exact integer.
CALCULATING TEAM COMPETITION
ICTM REGIONAL 2012 DIVISION AA
PAGE 3 of 3
14.
Sheila, the seamstress, buys her cloth in strips that are 30 inches wide and 864 inches
long. She estimates that in making a shirt there is a waste amount of 8.96% of each 30inch wide rectangular piece of cloth that she cuts from her long strip. If the finished shirt
actually contains 752 square inches of cloth, find the number of shirts that Sheila can
make from one 30 inch wide by 864 inch strip of cloth. Round your answer to the nearest
integer and express your answer as that integer.
15.
Find the amplitude of the graph of y  3.459sin( )  7.248cos( ) .
16.
The radius of the wheel of an exercise bike is 10 inches. If the wheel is turning 3 times
per second, determine the linear speed in mph. of a point on the circumference of the
wheel.
17.
The length of a slant height of a regular square pyramid is 65. The length of a side of the
square base is an even integer, and the length of the altitude is an integer. Find the
absolute value of the difference between the largest and the smallest possible volume of
such a pyramid. Express your answer as an exact integer.
18.
Find the slope of a line that passes through 13.17, 19.25  and  68.86,1728.1 .
19.
Assume that in the world’s general population, 90% of the persons are predominantly
right-handed, 9% of the persons are predominantly left-handed, and 1% are
ambidextrous. If Jim, Bill, and John are 3 persons selected at random from the world’s
general population, find the probability that at least 2 of those 3 persons are
predominantly left-handed. Express your answer as a decimal rounded to the nearest
thousandth. Do not use scientific notation.
20.
A particle must travel from (6,  5,10) to a point R on the plane whose equation is
x  2 y  2 z  9 , and then from point R to the point (3,  8,13) . The particle travels at
constant rates of 3 units per second from (6,  5,10) to point R and 4 units per second
from point R to (3,  8,13) . Find the minimum possible number of seconds necessary to
complete the required journey. Express your answer as a decimal rounded to the
nearest hundred-thousandth. Do not use scientific notation.
2012 RAA
School
Calculator Team
Correct X
ANSWERS
(Use full school name – no abbreviations)
5 pts. ea. =
Note: All answers must be written legibly. Round answers to four significant digits and write
in either standard or scientific notation unless otherwise specified in the question. Except
where noted, angles are in radians. No units of measurement are required.
1.
2.
3.
4.
5.
6.
7.
1129 OR 1.129  103
187.1 OR 1.871 10
2
7.691 OR 7.691 10
195.3 OR
1.953  10 2
13338'
11.
12.
0
13.
(Degrees
optional.)
14.
(Must be in degree
and minutes form.)
45386.692
15.
(Must be this
decimal.)
16.
3.397 OR 3.397  10
3362
0
(Must be this
integer.)
8.
9.
10.
17.
18.
1192.325
1568
(Must be this
decimal.)
19.
(Must be this
integer.)
20.
1172 OR 1.172  103
8
3
(Must be this reduced
improper fraction.)
372
31
(Must be this
integer.)
(Must be this
integer.)
8.031 OR 8.031 100
10.71 OR 1.071 10 (mph.
optional.)
OR 1.071 101
119104
(Must be this
integer )
31.38 OR 3.138  10
OR 3.138  101
0.023 OR .023
6.11873
(Must be this
decimal.)
(Must be this
decimal.)
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2012 REGIONAL DIVISION AA
1.
Find the value of x such that 4
2 x 5
PAGE 1 OF 2

1
. Express your answer as a decimal.
8
A
2.
In the figure with degree measures as shown, let k be
the degree measure of ABC . The roots of
(5 x  4)2  ( x  3) 2  3 are equivalent to the roots of
B
83
58
ax 2  17 x  b  0 . Find the value of  a  b  k  .
93
143
E
3.
Find the sum of the distinct roots of the following four equations:
5 x  8  112
4.
D
y 2  4 y  12  0
z 2  288
x  1  10
In ABC with A 1, 5  , B  3, 7  , and C  4, 4  , find the length of the median from C
to AB .
5.
Working alone at constant rates , it takes the respective number of hours to clean Room
379: Karen, 3; Kay, 7; Tom, 13. With no loss of efficiency, let k and w be the respective
number of hours required if the first two work together and then if the last two work
together. Expressed as a decimal, find the value of  k  w  .
6.
Let x be an integer such that 80  x  89 . Let S be the sum of all possible distinct
values of x such that it is impossible to use exactly x American coins that will have a
monetary value of one dollar. Find the value of S .
7.
An equilateral quadrilateral has diagonals whose lengths are 56 and 90. Find the
perimeter of the equilateral quadrilateral.
8.
Let y vary inversely as x . If y  3 when x  2 , find the value of y when x  5 . Let q
vary jointly as z and h . If q  12 when h  3 and z  2 , find the value of q when
z  5 and h  4 . For your answer, give the value of  y  q  as a decimal.
9.
Some right triangles with three integer side lengths satisfy the property that area and
perimeter are numerically equal. Find the sum of the perimeters of all such noncongruent triangles.
C
FROSH-SOPH 2 PERSON COMPETITION
ICTM 2012 REGIONAL DIVISION AA
10.
Players stand in a circle. Player 1 stays in. Player 2 is knocked out. Player 3, in; Player
4, out. This continues, knocking every other Player out, until only one Player remains.
With 9 Players, let k be the number of the last Player remaining; with 11 players, let w
be the number of the last Player remaining. Find the value of  2k  3w  .
ANSWERS
1.
3.25 (Must be this decimal.)
2.
177
3.
30
4.
PAGE 2 OF 2
34
5.
6.65 (Must be this decimal.)
6.
341
7.
212
8.
41.2 (Must be this decimal.)
9.
54
10.
27
2012 RAA
School
Fr/So 2 Person Team
ANSWERS
(Use full school name – no abbreviations)
NOTE: Questions 1-5 only
are NO CALCULATOR
Total Score (see below*) =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
Answer
Score
(to be filled in by proctor)
1.
3.25
2.
177
3.
30
4.
5.
34
6.65
8.
341
212
41.2
9.
54
10.
27
6.
7.
(Must be this decimal.)
(Must be this decimal.)
(Must be this decimal.)
TOTAL SCORE:
(*enter in box above)
Extra Questions:
11.
12.
13.
39.5
32
95
14.
60
15.
2
(Must be this decimal,
degrees optional.)
* Scoring rules:
Correct in 1st minute – 6 points
Correct in 2nd minute – 4 points
Correct in 3rd minute – 3 points
PLUS: 2 point bonus for being first
In round with correct answer
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2012 REGIONAL DIVISION AA
1.
PAGE 1 OF 2
k 7 1
Let 0
2 3  152 . Let w be the area of a triangle with vertices at  0,0  ,  0,10  ,
4 0 6
and  4,3 . Find the value of  k  w  .
2.
Let a  1 and use log a 8  5.7132 as an exact decimal in this problem. Find the value of
log a 2 . Express your answer as an exact decimal.
3.
Find the smallest possible positive value for x for any point that lies on the conic whose
equation is
4.
5.
 x  3
16
2
y2

 1.
25
1 k 1
1 4 2
If 2 w 3  5 and 3 k 1  110 , find the value of  k  5w  .
3 2 1
5 w 5
A line passes through the foci of the parabolas whose equations are y 2  4 x and
x 2   8 y . If the equation of that line is expressed in the form y  mx  b , find the value
of  5m  3b  .
6.
Find the sum of 0.4 (where the “4” repeats) and 0.672 (where the “72” repeats).
Express your answer as an improper fraction reduced to lowest terms.
7.
If 90  x  720 , let S be the sum of all distinct values of x such that cos( x)  sin( x) .
Let k be the sum of the terms of an infinite geometric progression for which the first
1
term is 784 and the common ratio is . Find the value of  S  k  .
3
8.
Let k be the tangent of the smallest angle of a right triangle whose hypotenuse has a
length of 89 and one of whose legs has a length of 39.  x  is defined as the greatest
integer which is not greater than x . If 2  x  3 , let w be the largest possible integral
2
value of  x 2    x  . Find the value of 1600k  w  .
9.
Let f be a function for which f (1)  12 , f (3)  5 , and f (k )  3  f (k  1)   f (k  1) for
all positive integral k . Let  x, y  be the focus of the parabola whose equation is
y
1 2
x  16 x  1 . Find the value of  x  y  f  4   .
8
JUNIOR-SENIOR 2 PERSON COMPETITION
ICTM 2012 REGIONAL DIVISION AA
10.
PAGE 2 OF 2 By substituting 1, 2, 3, 4, and 5 for x in P  x  , the first five values for P  x  are
respectively 5, 18, 35, 56, and 81. By substituting 1, 2, 3, 4, and 5 for x in R  x  , the
first five values for R  x  are respectively 3, 17, 33, 51, and 71. If P  x  and R  x  are
polynomial expressions of lowest degree satisfying the given, find P  25   R  25  .
ANSWERS
1.
25
2.
1.9044 (Must be this decimal.)
3.
7
4.
64
5.
4
6.
553
(Must be this reduced improper fraction.)
495
7.
2391
8.
784
9.
691
10.
2312
2012 RAA
School
Jr/Sr 2 Person Team
ANSWERS
(Use full school name – no abbreviations)
NOTE: Questions 1-5 only
are NO CALCULATOR
Total Score (see below*) =
Note: All answers must be written legibly in simplest form, according to the specifications
stated in the Contest Manual. Exact answers are to be given unless otherwise
specified in the question. No units of measurement are required.
Answer
Score
(to be filled in by proctor)
1.
2.
25
1.9044
3.
7
4.
64
5.
4
6.
553 (Must be this reduced improper fraction.)
495
8.
2391
784
9.
691
10.
2312
7.
(Must be this decimal.)
TOTAL SCORE:
(*enter in box above)
Extra Questions:
4
* Scoring rules:
11.
Correct in 1st minute – 6 points
12.
9
13.
14.
15.
 2, 8 (Must be this ordered pair.)
91
8
Correct in 2nd minute – 4 points
Correct in 3rd minute – 3 points
PLUS: 2 point bonus for being first
In round with correct answer
ICTM Regional Contest 2012
Div AA Oral Competition
Taxicab Geometry
The answers to all of the following problems can fit onto a single copy of the attached
coordinate grid. You will be provided with a single copy of the grid on a transparency
when you are presenting.
1.
In relation to A(– 4,8) and B(– 9,5), show graphically all points that are taxicab
equidistant from A and B. What name might you use to describe these points?
2.
Given S(– 10,– 3) and T(– 4,– 5), describe and sketch the set of points P,
P | dT SP   dT TP   12 .
3.
Ideal City’s School District 2 is represented as the 1st quadrant of the plane. The four
high schools in the district are located at K(4,8), L(10,4), M(12,10), and N(5,13), and
everyone in District 2 attends the school closest to their home. Heather lives at
H(10,11), but wants move so that her children will be in the part of the district served
by high school N and be within one block of the boundary of the school. If she also
wants a corner house (at a lattice point), what is the minimum number of blocks that
she will have to move? Where could her new home be located?
ICTM Regional Contest 2012
Div AA Oral Competition
Taxicab Geometry
Graph Paper
ICTM Regional Contest 2012
Div AA Oral Competition
Taxicab Geometry
Extemporaneous Questions
THIS SHEET SHOULD BE GIVEN TO STUDENTS AT THE BEGINNING OF THE
EXTEMPORANEOUS QUESTION PERIOD.
You may continue to use the transparency to answer these questions.
E1.
Given the points S  4,  2  and T 1,  6  , compute the Euclidean distance and the
taxicab distance between the two points.
E2.
With two given points E  x1 , y1  and F  x2 , y 2  , the Euclidean distance between the
points and the taxicab distance between the two points are equal. What must be true
about the relationship(s) between any or all of x1 , y1 , x2 , y 2 ?
E3.
Given point C (6, – 14), graphically show all points P such that the taxicab distance
CP is equal to 4 and all points R such that the Euclidean distance CR is equal to 4.
What taxicab and Euclidean shapes are described by these points?
ICTM Regional Contest 2012
1.
Div AA Oral Competition
Taxicab Geometry
SOLUTIONS
In relation to A(– 4,8) and B(– 9,5), show graphically all points that are taxicab
equidistant from A and B. What name might you use to describe these points?
All points equidistant from two points would form the taxicab midset or the
taxicab perpendicular bisector of AB .
See quadrant II graph.
2.
Given S(– 10,– 3) and T(– 4,– 5), describe and sketch the set of points P,
P | dT SP   dT TP   12 .
The shape is a taxicab ellipse, since the sum of the distances to the two fixed
points is constant. Students may note that the points S and T are the foci of the
ellipse. (Note: the center of the taxi-ellipse is at (- 7, - 4). Students do not need
to give this point; it is included for reference only.)
See quadrant III graph.
3.
Ideal City’s School District 2 is represented as the 1st quadrant of the plane. The four
high schools in the district are located at K(4,8), L(10,4), M(12,10), and N(5,13), and
everyone in District 2 attends the school closest to their home. Heather lives at
H(10,11), but wants move so that her children will be in the part of the district served
by high school N and be within one block of the boundary of the school. If she also
wants a corner house (at a lattice point), what is the minimum number of blocks that
she will have to move? Where could her new home be located?
3 blocks. Three different points are possible for the new home, as shown on the
graph, at (7, 11), (8, 12) and (9, 13).
Students should be given credit for finding any one of the possibilities and may
just point out the location(s) on the graph. They should not be penalized if they
do not include the coordinates of the new home.
Note: only the boundary between N and M is necessary to solve the problem, so
students should not be penalized for omitting the others.
See quadrant I graph.
ICTM Regional Contest 2012
Div AA Oral Competition
Taxicab Geometry
SOLUTIONS – Extemporaneous Questions
E1.
Given the points S  4,  2  and T 1,  6  , compute the Euclidean distance and the
taxicab distance between the two points.
Euclidean distance is 5, using distance formula or Pythagorean Theorem
( 32  42 ). The taxicab distance is 7 ( 3  4 ).
See quadrant IV graph.
E2.
With two given points E  x1 , y1  and F  x2 , y 2  , the Euclidean distance between the
points and the taxicab distance between the two points are equal. What must be true
about the relationship(s) between any or all of x1 , y1 , x2 , y 2 ?
If the distances are the same, the points must lie on a horizontal or vertical line.
Students may give either x1  x2 or y1  y 2 or both.
E3.
Given point C (6, – 14), graphically show all points P such that the taxicab distance
CP is equal to 4 and all points R such that the Euclidean distance CR is equal to 4.
What taxicab and Euclidean shapes are described by these points?
See quadrant IV graph. The graphs are a taxicab circle and a Euclidean circle.
SOLUTIONS and
NOTES FOR JUDGES
Problem #3:
minimum distance
from H is 3; the
three possibilities
are shown by the
“open” dots inside
the boundaries of
N.
Problem #1
Taxicab
perpendicular
bisector of segment
connecting A and B.
Only the
boundary of N
and M need to
be shown; the
others are for
reference
only.
N
H
M
K
A
B
L
Extemp #1 Euclidean
distance (solid segment)
is 5; Taxicab distance
(dashed segments) is 7.
Note: taxicab distance
could be either set of
dashed segments.
P
S
T
Q
Problem #2:
Taxicab ellipse
with foci S and T
C
Extemp #3
graph;
Taxicab circle
(all pts P) and
Euclidean circle
(all pts R)